名校
1 . 用数学归纳法证明“当
为正奇数时,
能被
整除”的第二步是:设
,则假设
=______ 时正确,再推
=______ 时正确.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c41c0c0df2d1dd2b1f065f1df228ad81.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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2 . 下列语句中是命题的有________ ;是真命题的有________ (填序号).
①这里真热闹啊!②求证
是无理数;③一个数不是正数就是负数;④并非所有的人都喜欢苹果;⑤若x=2,则
.
①这里真热闹啊!②求证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91ed0dd6412acd9fdc5a558bff75fb98.png)
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3 . 用数学归纳法证明
(
且
),第一步要证明的不等式是______ ,从
到
时,左端增加了________ 项.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6846743e55bb6f2ee46b2d03ba626461.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a70b95c53fb6655721e2a8c61f5c2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10e468312d09c6563c9094b710a35a65.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
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解题方法
4 . 正多面体也称柏拉图立体(被誉为最有规律的立体结构)是所有面都只由一种正多边形构成的多面体(各面都是全等的正多边形).数学家已经证明世界上只存在五种柏拉图立体,即正四面体、正六面体、正八面体、正十二面体、正二十面体.已知一个正八面体
的棱长都是2(如图),
、
分别为
、
的中点,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78aa2d427fb896e5e192c2032a62b81b.png)
______ .若
,过点
的直线分别交直线
于
两点,设
(其中
均为正数),则
的最小值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9165d9bfbb0f0d19eb482c2a4c1b29b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b88ea271b3352d75008832f129d39dc0.png)
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名校
5 . 古希腊数学家欧几里得所著《几何原本》中的“几何代数法”,很多代数公理、定理都能够通过图形实现证明,并称之为“无字证明“如图,
为线段
中点,
为
上的一点.以
为直径作半圆,过点
作
的垂线,交半圆于
.连结
,
,
,过点
作
的垂线,垂足为
.设
,
,则图中线段
,线段
,线段________
;由该图形可以得出
,
,
的大小关系为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/16/3143184d-bc71-40e6-a813-5901a2a2c546.png?resizew=210)
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2022-10-14更新
|
354次组卷
|
6卷引用:专题25 欧几里得
6 . 《几何原本》中的几何代数法是指以几何方法研究代数问题,这种方法是后世西方数学家处理问题的重要依据,通过这一原理,很多代数公理或定理都能够通过图形实现证明,也称之为无字证明.现有图形如图所示,
为线段
上的点,且
,
,
为
的中点,以
为直径作半圆.过点
作
的垂线交半圆于
,连接
,
,
,过点
作
的垂线,垂足为
,过点
作
的垂线
,使得
.该图形完成
的无字证明.图中线段__________ 的长度表示
,
的调和平均数
,线段______________ 的长度表示
,
的平方平均数
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce1132157a33c82610c2d5035493d024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73dc39d974f75278287bb409ae179ee9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10d204a0acae411ed144c3b790f54c69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/701f3b0e2bedfe5195443459072d798e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce1132157a33c82610c2d5035493d024.png)
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![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/20/19622bc6-51ec-4aa7-9b50-7ffc12f43a0d.png?resizew=190)
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7 . 用数学归纳法证明
(
为正整数)时,假设
成立,要证
时等式成立中的等式应为______ ;从
到
,等式左边需增加的代数式为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad47db6cbb534751610ed80fdbbf5832.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ef7ca2b3e8061384501f668e59696a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ef7ca2b3e8061384501f668e59696a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
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名校
解题方法
8 . 阿波罗尼斯的著作《圆锥曲线论》是古代世界光辉的科学成果,它将圆锥曲线的性质网罗殆尽几乎使后人没有插足的余地.他证明过这样一个命题:平面内与两定点距离的比为常数
(
且
)的点的轨迹是圆,后人将这个圆称为阿氏圆,现有
则
的面积最大值为______ ,此时AC的长为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f0d68648b10fce54dfc19c5ee60086d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c525393775354325cbf7839366ca50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0b3c1531df000ec0d6be8837fbdb4a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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9 . 阿波罗尼斯(约公元前262-190年)证明过这样一个命题:平面内到两定点距离之比为常数
的点的轨迹是圆,后人将此圆称为阿氏圆.若平面内两定点
、
间的距离为4,动点
满足
,则动点
的轨迹所围成的图形的面积为___________ ;
最大值是___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/533d0df0ab043fd32dce4c348c7b30e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91c27676db1dadac691fb5f50f6155a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a33e5d0dbdd0f15854f0d7dd8b53058.png)
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10 . 阿波罗尼斯(古希腊数学家,约公元前262-190年)的著作《圆锥曲线论》是古代世界光辉的科学成果,它将圆锥曲线的性质网罗殆尽,几乎使后人没有插足的余地.他证明过这样一个命题:平面内与两定点距离的比为常数
的点的轨迹是圆,后人将这个圆称为阿波罗尼斯圆.①若定点为
,写出
的一个阿波罗尼斯圆的标准方程__________ ;②△
中,
,则当△
面积的最大值为
时,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd707b69a11f8de5566f23c1a2a9ff5a.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b991d4173297923de7c4c1fa48bfae61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f01b186ac8aa73e1a3609b40b6c3ee6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3a2a34b4317deffa40ba34e269c2b81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
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2020-06-26更新
|
517次组卷
|
3卷引用:专题38 圆与方程-学会解题之高三数学万能解题模板【2022版】